Time Efficient Face Recognition Using Stable Gram-Schmidt ... - SERSC

International Journal of Signal Processing, Image Processing and Pattern Vol. 2, No.1, March, 2009

Time Efficient Face Recognition Using Stable Gram-Schmidt Orthonormalization I. Sajid, M.M. Ahmed and I. Taj Department of Electronic Engineering Mohammad Ali Jinnah University (MAJU), Jinnah Avenue, Islamabad {imtiazsajid, mansoor, imtiaztaj}@jinnah.edu.pk Abstract Most commonly used face recognition algorithms are based on extraction of global features using eigenvalue decomposition of some relational matrix of image intensity values. Real time face recognition applications require a computationally efficient algorithm for eigenvalues generation. Fast principal component analysis (FPCA) is an algorithm for efficient generation of eigenvalues which improves the computational efficiency to O(n2) as compared to normal decomposition method which gives the solution in O(n3) time. In FPCA however, nonconvergence state can be resulted for high resolution images because in this case the number of Grams-Schmidt (GS) iterations for orthonormalization convergence may exceed the maximum limit. To overcome this problem we present a modified FPCA algorithm to generate eigenvalues for images including those at high resolution. An overall efficient face recognition scheme has also been proposed using the generated eigenvalues, which can work satisfactorily under varying image resolutions. The validity of the proposed system has been checked by varying the feature vectors and the training sets. The developed technique provides an efficient and a low error rate solution for high speed image recognition systems.

1. Introduction Biometric based human verification and identification have found many applications in recent years ranging from security of personal money transactions to large scale security applications such as airport security. Centuries ago Babylonian kings used clay finger prints for authenticity [1]. Various biometrics have since been employed for personal verification and identification such as fingerprints, palmprints, iris patterns, face features, voice and human gait etc. Human face is one of the most famous and natural biometric, and face recognition systems are usually preferred over the other biometric systems due to their wider acceptability and ease of operation. However, contrary to the general perception, face recognition is not a straightforward problem and is vulnerable to many variations such as intensity variations, pose change, aging, occlusions etc. A variety of techniques have been proposed in the literature to model a human face in such a way that the final matching is immune to these variations [2-6]. These techniques can generally be classified into local feature extraction techniques and global features extraction techniques [3]. Local feature extraction techniques usually calculate autocorrelation matrix within each image in addition to the co-variance matrix between images [4], and they are computationally expensive. In global feature extraction techniques only covariance is computed, and thus they are relatively more efficient.

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In 1991, Turk and Pentland [2] proposed their famous approach to extract the most meaningful face details for matching, using principal component analysis (PCA). In their approach they mapped the face vector onto a smaller number of basis vectors called eigen faces corresponding the highest eigenvalues of the covariance matrix. They also presented a scheme to efficiently compute the eigenvalues by considering a matrix which is sufficiently smaller than the original covariance matrix. It is obvious that in their approach the timing efficiency of the recognition system directly depends on the number of highest eigenvalues but they did not suggest appropriate number of eigenvalues for a given face recognition application. Real time face recognition applications require an efficient system [3], thus appropriate selection of eigenvalues that minimizes the computational requirements and maximizes the system performance within constraints is vital. The PCA which was followed by Turk et al. [2] is one of the holistic algorithms [3-5] which is based on global feature extraction. In PCA these features are extracted by computing the eigenvalues normally by using decomposition (EVD) methods. EVD method requires tridiagonalization by taking co-variance symmetric matrix and then decomposition of tridiagonal matrix is accomplished. Therefore, operational time complexity of PCA, because of tridiagonalization and decomposition, becomes O(n3) [7, 8]. Decomposition method cannot be stopped for desired number of leading eigenvalues. However usually 3-4 leading eigenvalues occupy more than 90% weight of the total variance [9,10], whereas recognition accuracy depends on variance in the eigenspace. To improve the efficiency of PCA based systems, the concept of FPCA was introduced [7]. FPCA used Gram-Schmidt orthonormalization to calculate leading eigenvalues (LEVs) but with lesser time complexity i.e. O(n2) [7]. However, mean square error (MSE) in FPCA is higher [7] than PCA, for images of size greater than 4000 pixels. Moreover, face recognition using FPCA decreases the decision time of a system but could lead to a non-convergence state, especially when high resolution images are used [11]. Whereas images may vary from as few as 256 pixels to 452736 pixels [2,6,11], according to the resolution used. In other words, FPCA based system is affected by non-convergence state of the algorithm and high MSE. For a deterministic state, the system should identify the parameters or characteristics which control the convergence condition. We recently established that a definite convergence can be attained, for high resolution images, with modified Grams-Schmidt (GS) process used in FPCA [11]. The technique, called adaptive FPCA (AFPCA) has the capability to adjust minimum possible tolerance value according to image resolution and make the real-time face recognition deterministic and time efficient. Accuracy, decision and learning time of AFPCA increases by increasing the number of principal components (h), images in the training set (TS) and image resolutions in TS respectively whereas a face recognition system tries to achieve maximum possible accuracy within minimum given time. Thus appropriate selection of h, TS and images resolution provides efficient face recognition than conventional PCA based system. Selection of h is not a straight forward decision [12]. The proposed technique also minimizes error rate in face recognition, but at the expense of time. This paper, focuses only on the time efficient aspect of a face recognition system based on AFPCA.

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2. Limitations of FPCA It is observed that both PCA and FPCA generates similar MSE, if the chosen value of ε = 0.01 which has been suggested in [7], under the following conditions: i. At least six samples are used in TS. ii. Features vector (d) of orthogonal space should be equal to 100. iii. The value of h should be 10. iv. Image size should be less than 4000 pixels. The maximum probability of convergence of FPCA is observed for multi pose images having resolution 118 x 122, d=100 and h=10. Variation of six faces with different facial expression is treated as multi pose. The concept of multi pose is used in different experiments as shown in Table-1. The number of iterations depends heavily on the image resolution. It has been demonstrated that the technique is valid even for the images which have been acquired with various lighting effects. Non-convergence in FPCA is addressed when iterations in GS process exceed 1000. These non-convergence conditions are highlighted with shaded cells in Table-1. The convergence criterion in FPCA is based on the rule that dot product of orthogonal vectors, which are generated by GS process, should be unity [13]. This product value is T

+ calculated with current Φ p and previous Φ p base vectors and its difference from unity

is computed as

Φ

+T

p

.Φ

p

−1

(1 )

Table 1. Limitation of FPCA when ε = 0.01, varying dimensionalities whereas shaded cells show non-convergence conditions. Image Size 66 x 59 66 x 59 66 x 59 66 x 59 66 x 59 118x 122 118x 122 118x 122 118x 122

h 10 10 10 10 10 10 10 10 10

Faces / Objec Faces Faces Faces Faces Object Faces Faces Faces Object

Single / Multi pose Multi Multi Single Three Single Muti Single Three Single

Epsilon 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

Max. Iterations 10 > 1000 > 1000 > 1000 11 10 > 1000 > 1000 > 1000

Training Images all 10 10,20 10 10,20 All 10,20 10,20 all

For convergence condition, the value of Equation (1) should be less than the chosen value ε. It is observed that when variation in poses are less than six in the TS having less than 100 images then Equation (1) approaches slowly to ε. This slow convergence is because of the stochastic and iterative nature of GS process. In case of non-convergence, GS multipies the co-variance with the modified vector Φ p and then check the convergence condition. Whereas the initial values of Φ p depend on the system random T generation process. GS process tries to orthogonalize the current Φ +p and previous Φ p

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vectors until it is less than ε which defines tolerance of the system. Ideally this tolerance should be zero, but practically the value of ε varies according to the randomness of Φ p in FPCA and rounding effect in all sorts of other iterative processes [14]. It is suggested that an adaptive value of ε will facilitate a definite convergence in a GS process. For this purpose, a modification in the existing algorithm is proposed. The MSE of the modified algorithm (AFPCA) is studied. AFPCA ensures avoidance infinite looping by adjusting value of ε, or by deciding the maximum number of iterations.

3. Modification in FPCA algorithm Transformation is of dimension d x h

Φ:a → b a should be zero mean where a ∈ R d (d − dim ensional space) & b ∈ R h (reduced h − dim ensional space) step i. Select h, the number of leading principal components and compute co-variance

set p ← 1 step ii. Eigenvector Φ p of size d x 1 is initialized randomly and initialize matrix vv of size d x h and set converge ← 1. a. Set ε ← 0.01 and initialize iteration counter (IC) with zero for post-condition ∑x

step iii. Assign Φ p as Φ p ← ∑ x Φ p step iv. Loop the Gram- Schmidt orthogonalization process

(

p −1

)

+ T step v. Φ p ← Φ p − ∑ Φ p .Φ j .Φ j j =1

+ step vi. Normalize Φ +p by dividing it by its norm Φ p ←

step vii. Compute the convergence condition as Φ

+T

p

Φ +p Φ +p

.Φ

p

−1